Is there an analytical method of looking at the geometric mean that will allow one to break it down to its various components?
The focus of the question is more for financially related returns, but I am open to consider other fields as well to see if it is applicable.
So the aim of the question is to get some sort of results along the lines of
GEOMETRIC MEAN = f(ARITHMETIC MEAN,X ,Y,Z)
where $X,Y,Z$ are all factors/variables that help determine the value of the geometric mean.
I know there is a difference in the way its calculated etc, but I guess what I am looking for is what specific factor drives the difference between arithmetic and geometric mean. Most of the time, in financially related contexts arithmetic mean tends to be larger in magnitude than geometric mean due to the variability of return values, i.e. where $x_i$ does not equal $x_j$ where the arithmetic mean is the $\sum \limits_{i=1}^n\frac{x_i}{n}$, and the geometric mean is $\prod \limits_{i}^n {(x_i+1)}^{\frac{1}{n}} - 1$, but just saying that all the returns are not the same is what causes the difference to not quite feel exact enough.
Is there something that is better that will show me exactly what causes the gap between geometric and arithmetic means?
Approximations using Taylor series would be ok too...